Problem: Find the distance between the point ${(-3, -3)}$ and the line $\enspace {x = 3}\thinspace$. {1} {2} {3} {4} {5} {6} {7} {8} {9} {\llap{-}2} {\llap{-}3} {\llap{-}4} {\llap{-}5} {\llap{-}6} {\llap{-}7} {\llap{-}8} {\llap{-}9} {1} {2} {3} {4} {5} {6} {7} {8} {9} {\llap{-}2} {\llap{-}3} {\llap{-}4} {\llap{-}5} {\llap{-}6} {\llap{-}7} {\llap{-}8} {\llap{-}9}
Explanation: First, find the equation of the perpendicular line that passes through ${(-3, -3)}$ Since the blue line has an infinite slope, the perpendicular line will have a slope of ${0}$ and therefore will be a horizontal line. The equation of the perpendicular line that passes through ${(-3, -3)}$ is $\enspace {y = -3}\thinspace$ We can see from the graph that the two lines intersect at the point ${(3, -3)}$ . Thus, the distance we're looking for is the distance between the two red points. Since their $y$ components are the same, the distance between the two points is simply the change in $x$ $|{-3} - ( {3} )| = 6$ The distance between the point ${(-3, -3)}$ and the line $\enspace {x = 3}\enspace$ is $\thinspace6$.